Multivariate distributions of clinical covariates at the time of cancer detection

Novel predictions of invasive breast cancer risk in mammography screening cohorts

Mammography screening programs are aimed at reducing mortality due to breast cancer by detecting tumors at an early stage. There is currently interest in moving away from the age-based screening programs, and toward personalized screening based on individual risk factors. To accomplish this, risk prediction models for breast cancer are needed to determine who should be screened, and when. We develop a novel approach using a (random effects) continuous growth model, which we apply to a large population-based, Swedish screening cohort. Unlike existing breast cancer prediction models, this approach explicitly incorporates each woman's individual screening visits in the prediction. It jointly models invasive breast cancer tumor onset, tumor growth rate, symptomatic detection rate, and screening sensitivity. In addition to predicting the overall risk of invasive breast cancer, this model can make separate predictions regarding specific tumor sizes, and the mode of detection (eg, detected at screening, or through symptoms between screenings). It can also predict how these risks change depending on whether or not a woman will attend her next screening. In our study, we predict, given a future diagnosis, that the probability of having a tumor less than (as opposed to greater than) 10-mm diameter, at detection, will be, on average, 2.6 times higher if a woman in the cohort attends their next screening. This indicates that the model can be used to evaluate the short-term benefit of screening attendance, at an individual level.

A unifying framework for continuous tumour growth modelling of breast cancer screening data

The aim of the current article is to present theory that can help unify continuous growth approaches for modelling breast cancer tumour growth based on human data. We present a framework that has three main features: a general likelihood function to account for patient specific screening regiments; stable disease assumptions describing tumour population dynamics; and mathematical models describing tumour growth, individual variation in tumour growth, a hazard for symptomatic detection, and screening test sensitivity. The framework is able to incorporate any random effects distributions for the tumour growth rate parameter, any hazard functions for symptomatic tumour detection, as well as any monotonously increasing function for the tumour growth model. Based on a sample of 1902 incident breast cancer cases with data on mammography screening, we show how the framework can be used to estimate tumour growth based on different growth functions.

Estimating Distributions of Breast Cancer Onset and Growth in a Swedish Mammography Screening Cohort

BACKGROUND

In recent years, biologically motivated continuous tumor growth models have been introduced for breast cancer screening data. These provide a novel framework from which mammography screening effectiveness can be studied.

METHODS

We use a newly developed natural history model, which is unique in that it includes a carcinogenesis model for tumor onset, to analyze data from a large Swedish mammography cohort consisting of 65,536 participants, followed for periods of up to 6.5 years. Using patient data on age at diagnosis, tumor size, and mode of detection, as well as screening histories, we estimate distributions of patient's age at onset, (inverse) tumor growth rates, symptomatic detection rates, and screening sensitivities. We also allow the growth rate distribution to depend on the age at onset.

RESULTS

We estimate that by the age of 75, 13.4% of women have experienced onset. On the basis of a model that accounts for the role of mammographic density in screening sensitivity, we estimated median tumor doubling times of 167 days for tumors with onset occurring at age 40, and 207 days for tumors with onset occurring at age 60.

CONCLUSIONS

With breast cancer natural history models and population screening data, we can estimate latent processes of tumor onset, tumor growth, and mammography screening sensitivity. We can also study the relationship between the age at onset and tumor growth rates.

IMPACT

Quantifying the underlying processes of breast cancer progression is important in the era of individualized screening.

Lymph node metastases in breast cancer: Investigating associations with tumor characteristics, molecular subtypes and polygenic risk score using a continuous growth model

We investigate the association between rate of breast cancer lymph node spread and grade, estrogen receptor (ER) status, progesteron receptor status, decision tree derived PAM50 molecular subtype and a polygenic risk score (PRS), using data on 10 950 women included from two different data sources. Lymph node spread was analyzed using a novel continuous tumor progression model that adjusts for tumor volume in a biologically motivated way and that incorporates covariates of interest. Grades 2 and 3 tumors, respectively, were associated with 1.63 and 2.17 times faster rates of lymph node spread than Grade 1 tumors (P < 10^-16 ). ER/PR negative breast cancer was associated with a 1.25/1.19 times faster spread than ER/PR positive breast cancer, respectively (P = .0011 and .0012). Among the molecular subtypes luminal A, luminal B, Her2-enriched and basal-like, Her2-enriched breast cancer was associated with 1.53 times faster spread than luminal A cancer (P = .00072). PRS was not associated with the rate of lymph node spread. Continuous growth models are useful for quantifying associations between lymph node spread and tumor characteristics. These may be useful for building realistic progression models for microsimulation studies used to design individualized screening programs.

Random effects models of lymph node metastases in breast cancer: quantifying the roles of covariates and screening using a continuous growth model

We recently described a joint model of breast cancer tumor size and number of affected lymph nodes, which conditions on screening history, mammographic density, and mode of detection, and can be used to infer growth rates, time to symptomatic detection, screening sensitivity, and rates of lymph node spread. The model of lymph node spread can be estimated in isolation from measurements of tumor volume and number of affected lymph nodes, giving inference identical to the joint model. Here, we extend our model to include covariate effects. We also derive theoretical results in order to study the role of screening on lymph node metastases at diagnosis. We analyze the association between hormone replacement therapy (HRT) and breast cancer lymph node spread, using data from a case-control study designed specifically to study the effects of HRT on breast cancer. Using our method, we estimate that women using HRT at time of diagnosis have a 36% lower rate of lymph node spread than nonusers (95% confidence interval [CI] =(8%,58%)). This can be contrasted with the effect of HRT on the tumor growth rate, estimated here to be 15% slower in HRT users (95% CI = (-34%,+7%)). For screen-detected cancers, we illustrate how lead time can relate to lymph node spread; and using symptomatic cancers, we illustrate the potential consequences of false negative screens in terms of lymph node spread.

Statistical models of tumour onset and growth for modern breast cancer screening cohorts

Historically, multi-state Markov models have been used to study breast cancer incidence and mammography screening effectiveness. In recent years, more biologically motivated continuous tumour growth models have emerged as alternatives. However, a number of challenges remain for these models to make use of the wealth of information available in large mammography cohort data. In particular, methodology is needed to address random left truncation and individual, asynchronous screening. We present a comprehensive continuous random effects model for the natural history of breast cancer. It models the unobservable processes of tumour onset, tumour growth, screening sensitivity, and symptomatic detection. We show how the unknown model parameter values can be jointly estimated using a prospective cohort with diagnostic data on age and tumour size at diagnosis, and individual screening histories. We also present a microsimulation study calibrated to population breast cancer incidence data, and to data on mode of detection and tumour size. We highlight the importance of adjusting for random left truncation, derive tumour doubling time distributions for screen-detected and interval cancers, and present results concerning the relationship between tumour presence time and age at diagnosis.

Joint models of tumour size and lymph node spread for incident breast cancer cases in the presence of screening

Continuous growth models show great potential for analysing cancer screening data. We recently described such a model for studying breast cancer tumour growth based on modelling tumour size at diagnosis, as a function of screening history, detection mode, and relevant patient characteristics. In this article, we describe how the approach can be extended to jointly model tumour size and number of lymph node metastases at diagnosis. We propose a new class of lymph node spread models which are biologically motivated and describe how they can be extended to incorporate random effects to allow for heterogeneity in underlying rates of spread. Our final model provides a dramatically better fit to empirical data on 1860 incident breast cancer cases than models in current use. We validate our lymph node spread model on an independent data set consisting of 3961 women diagnosed with invasive breast cancer.

Modelling breast cancer tumour growth for a stable disease population

Statistical models of breast cancer tumour progression have been used to further our knowledge of the natural history of breast cancer, to evaluate mammography screening in terms of mortality, to estimate overdiagnosis, and to estimate the impact of lead-time bias when comparing survival times between screen detected cancers and cancers found outside of screening programs. Multi-state Markov models have been widely used, but several research groups have proposed other modelling frameworks based on specifying an underlying biological continuous tumour growth process. These continuous models offer some advantages over multi-state models and have been used, for example, to quantify screening sensitivity in terms of mammographic density, and to quantify the effect of body size covariates on tumour growth and time to symptomatic detection. As of yet, however, the continuous tumour growth models are not sufficiently developed and require extensive computing to obtain parameter estimates. In this article, we provide a detailed description of the underlying assumptions of the continuous tumour growth model, derive new theoretical results for the model, and show how these results may help the development of this modelling framework. In illustrating the approach, we develop a model for mammography screening sensitivity, using a sample of 1901 post-menopausal women diagnosed with invasive breast cancer.

Optimal screening schedules for prevention of metastatic cancer

We develop methodological, mathematical, statistical, and computational approaches to constructing schedules of cancer screening that maximize the probability that by the time of primary tumor detection it has not yet metastasized. Solving this problem is based on a comprehensive mechanistic model of cancer progression. We apply the model with realistic parameters and the screening optimization methodology to mammographic screening for breast cancer within the American female population. We uncover some general patterns of optimal screening schedules. We show that optimization of screening regimens leads to a significant reduction in the probability of detecting breast cancer that has already disseminated.

Tumor control probability in radiation treatment

Patients undergoing radiation therapy (and their physicians alike) are concerned with the probability of cure (long-term recurrence-free survival, meaning the absence of a detectable or symptomatic tumor). This is not what current practice categorizes as "tumor control (TC);" instead, TC is taken to mean the extinction of clonogenic tumor cells at the end of treatment, a sufficient but not necessary condition for cure. In this review, we argue that TC thus defined has significant deficiencies. Most importantly, (1) it is an unobservable event and (2) elimination of all malignant clonogenic cells is, in some cases, unnecessary. In effect, within the existing biomedical paradigm, centered on the evolution of clonogenic malignant cells, full information about the long-term treatment outcome is contained in the distribution Pm(T) of the number of malignant cells m that remain clonogenic at the end of treatment and the birth and death rates of surviving tumor cells after treatment. Accordingly, plausible definitions of tumor control are invariably traceable to Pm(T). Many primary cancers, such as breast and prostate cancer, are not lethal per se; they kill through metastases. Therefore, an object of tumor control in such cases should be the prevention of metastatic spread of the disease. Our claim, accordingly, is that improvements in radiation therapy outcomes require a twofold approach: (a) Establish a link between survival time, where the events of interest are local recurrence or distant (metastatic) failure (cancer-free survival) or death (cancer-specific survival), and the distribution Pm(T) and (b) link Pm(T) to treatment planning (modality, total dose, and schedule of radiation) and tumor-specific parameters (initial number of clonogens, birth and spontaneous death rates during the treatment period, and parameters of the dose-response function). The biomedical, mathematical, and practical aspects of implementing this program are discussed.

Why victory in the war on cancer remains elusive: biomedical hypotheses and mathematical models

We discuss philosophical, methodological, and biomedical grounds for the traditional paradigm of cancer and some of its critical flaws. We also review some potentially fruitful approaches to understanding cancer and its treatment. This includes the new paradigm of cancer that was developed over the last 15 years by Michael Retsky, Michael Baum, Romano Demicheli, Isaac Gukas, William Hrushesky and their colleagues on the basis of earlier pioneering work of Bernard Fisher and Judah Folkman. Next, we highlight the unique and pivotal role of mathematical modeling in testing biomedical hypotheses about the natural history of cancer and the effects of its treatment, elaborate on model selection criteria, and mention some methodological pitfalls. Finally, we describe a specific mathematical model of cancer progression that supports all the main postulates of the new paradigm of cancer when applied to the natural history of a particular breast cancer patient and fit to the observables.

Identifiability of the joint distribution of age and tumor size at detection in the presence of screening

In recent years, a stochastic model of cancer development and detection allowing for arbitrary screening schedules has been developed and applied to analysis of screening trials and population-based cancer incidence and mortality data. The model is entirely mechanistic, builds on a minimal set of biologically plausible assumptions, and yields the joint distribution of tumor size and age of a patient at the time of diagnosis. Whether or not parameters of the model can be estimated from data generated by cohort studies depends on model identifiability. The present paper provides a proof of this important property of the model.

Chapter 10: The University of Rochester Model of Breast Cancer Detection and Survival

This paper presents a biologically motivated model of breast cancer development and detection allowing for arbitrary screening schedules and the effects of clinical covariates recorded at the time of diagnosis on posttreatment survival. Biologically meaningful parameters of the model are estimated by the method of maximum likelihood from the data on age and tumor size at detection that resulted from two randomized trials known as the Canadian National Breast Screening Studies. When properly calibrated, the model provides a good description of the U.S. national trends in breast cancer incidence and mortality. The model was validated by predicting some quantitative characteristics obtained from the Surveillance, Epidemiology, and End Results data. In particular, the model provides an excellent prediction of the size-specific age-adjusted incidence of invasive breast cancer as a function of calendar time for 1975-1999. Predictive properties of the model are also illustrated with an application to the dynamics of age-specific incidence and stage-specific age-adjusted incidence over 1975-1999.